Pervasive nonlinear vibrations due to rod-obstacle contact

Abstract

Simple mechanical contact between a deformable body and a rigid obstacle can lead to rich nonlinear behavior even when the equations governing the body’s motion are otherwise linear. We study the nonlinear dynamics of contact both numerically and analytically in a model problem of an Euler–Bernoulli beam and a flat substrate. This problem is relevant to microelectromechanical systems (MEMS) and marine structures. We show the nonlinearity to be dominantly quadratic, thereby giving rise to superharmonic and combination resonances of order two. We then demonstrate that similar nonlinear dynamics can in fact arise in far more general circumstances with nonlinear rods in contact with curved surfaces.

A Appendix: finite element method

Here, we describe the numerical method used in Sect. 2 to solve Eqs. (2.1) to (2.5). A similar approach for a more general class of problems has been developed by Humer et al. [12, 13], motivated by the earlier method of Vu-Quoc and Li [29]. The first step is to introduce the stretching transformation $z=\xi /\gamma (t)$, which maps the variable spatial domain $[0,\gamma (t)]$ to the fixed one [0, 1]. By writing $y=y(\xi ,t)={\tilde{y}}(\gamma (t)\xi ,t)$ and then applying the chain rule, Eq. (2.1) becomes

$$\begin{aligned} \begin{aligned}&\rho _0 \bigg [\frac{\partial ^{2}{\tilde{y}}}{{\partial }t^{2}} +\left( \frac{2\dot{\gamma }^2}{\gamma }-\ddot{\gamma }\right) \frac{z}{\gamma } \frac{\partial \tilde{y}}{{\partial }z}-\frac{2\dot{\gamma }z}{\gamma } \frac{\partial ^{2}{\tilde{y}}}{{\partial }t{\partial }z}\\&\quad +\left( \frac{\dot{\gamma }z}{\gamma }\right) ^2 \frac{\partial ^{2}{\tilde{y}}}{{\partial }z^{2}}\bigg ] +\frac{1}{\gamma ^4} \frac{\partial ^{2}}{{\partial }z^{2}} \left( EI\frac{\partial ^{2}{\tilde{y}}}{{\partial }z^{2}}\right) =-\rho _0g. \end{aligned} \end{aligned}$$

(A.1)

The appropriate boundary conditions are ${\tilde{y}}=a$ and $\partial {\tilde{y}}/\partial z =0$ at $z=0$, and ${\tilde{y}}=\partial {\tilde{y}}/\partial z=\partial ^2{\tilde{y}}/\partial z^2=0$ at $z=1$. We have now transformed the free-boundary-value problem to one on a fixed domain, a simplification that comes at the cost of a substantially more complicated differential equation, Eq. (A.1), that contains an advective term as well as $\gamma$, ${\dot{\gamma }}$, and $\ddot{\gamma }$. Our method is quasi-Eulerian in the sense that fixed values of the coordinate z correspond neither to fixed material points nor to fixed locations in space.

In order to obtain the weak form for a typical interior element $\varOmega ^e=(z_1^e,z_2^e)\subset (0,1)$, we multiply Eq. (A.1) by the test function $w=w(z,t)$ and integrate the result over $\varOmega ^e$. After applying integration by parts twice to the bending stiffness term, we obtain the following:

$$\begin{aligned} \begin{aligned}&\int _{\varOmega ^e}w\rho _0 \frac{\partial ^{2}{\tilde{y}}}{{\partial }t^{2}}\,\mathrm {d}z+\left[ 2\left( \frac{\dot{\gamma }}{\gamma }\right) ^2 -\ddot{\gamma }\right] \int _{\varOmega ^e}w\rho _0z \frac{{\partial }\tilde{y}}{{\partial }z}\,\mathrm {d}z\\&\quad -\frac{2\dot{\gamma }}{\gamma }\int _{\varOmega ^e}w\rho _0z \frac{\partial ^{2}{\tilde{y}}}{{\partial }z{\partial }t}\,\mathrm {d}z +\left( \frac{\dot{\gamma }}{\gamma }\right) ^2\int _{\varOmega ^e}w\rho _0z \frac{\partial ^{2}{\tilde{y}}}{{\partial }z^{2}}\,\mathrm {d}z\\&\quad +\frac{1}{\gamma ^4}\int _{\varOmega ^e} \frac{\partial ^{2}w}{{\partial }z^{2}}EI \frac{\partial ^{2}{\tilde{y}}}{{\partial }z^{2}}\,\mathrm {d}z +\int _{\varOmega ^e}w\rho _0g\,\mathrm {d}z\\&\quad +\left[ w\frac{\partial }{{\partial }z} \left( EI\frac{\partial ^{2}{\tilde{y}}}{{\partial }z^{2}}\right) - \frac{{\partial }w}{{\partial }z}EI \frac{{\partial }^{2}\tilde{y}}{{\partial }z^{2}}\right] _{z_1^e}^{z_2^e}=0, \end{aligned} \end{aligned}$$

(A.2)

which can be readily discretized in the spatial dimension. We define the vector of element-wise generalized displacements to be

$$\begin{aligned}&[\mathbf {q}^e(t)]=\left[ \tilde{y}(z_1^e,t),\, \frac{{\partial }\tilde{y}}{{\partial }z}(z_1^e,t),\, \tilde{y}(z_2^e,t),\,\right. \nonumber \\&\quad \left. \frac{{\partial }\tilde{y}}{{\partial }z}(z_2^e,t)\right] ^T, \end{aligned}$$

(A.3)

and the element test function vector $[{\mathbf {w}}^e(t)]$ analogously. Introducing the normalized coordinate $s=(z-z_1^e)/h^e$, we use the following shape functions to interpolate ${\tilde{y}}$ and w:

$$\begin{aligned} N_1^e(s)&= 1-3s^2+2s^3, \end{aligned}$$

(A.4a)

$$\begin{aligned} N_2^e(s)&= h^es(1-s)^2, \end{aligned}$$

(A.4b)

$$\begin{aligned} N_3^e(s)&= s^2(3-2s), \end{aligned}$$

(A.4c)

$$\begin{aligned} N_4^e(s)&= h^es^2(s-1). \end{aligned}$$

(A.4d)

Writing the shape functions as the row vector $[{\mathbf {N}}^e(s)]=[N_1^e(s)\,\,N_2^e(s)\,\,N_3^e(s)\,\,N_4^e(s)]$, the interpolations are ${\tilde{y}}=[{\mathbf {N}}^e][{\mathbf {q}}^e]$ and $w=[{\mathbf {N}}^e][{\mathbf {w}}^e]$.

After assembly and imposition of the z-domain versions of the first four boundary conditions in Eq. (2.3), Eq. (A.2) gives rise to a global matrix problem of the form

$$\begin{aligned} \begin{aligned}&[{\mathbf {M}}][\ddot{{\mathbf {q}}}]-\frac{2{\dot{\gamma }}}{\gamma }[{\mathbf {D}}_1] [\dot{{\mathbf {q}}}]+\bigg \{\bigg (2\bigg (\frac{{\dot{\gamma }}}{\gamma }\bigg )^2-\frac{\ddot{\gamma }}{\gamma }\bigg )[{\mathbf {D}}_1]\\&\quad +\bigg (\frac{{\dot{\gamma }}}{\gamma }\bigg )^2[{\mathbf {D}}_2]+\frac{1}{\gamma ^4} [{\mathbf {D}}_3]\bigg \}[{\mathbf {q}}]=[{F(\gamma ,{\dot{\gamma }},\ddot{\gamma })}]. \end{aligned} \end{aligned}$$

(A.5)

The corresponding element matrices, which are apparent from Eq. (A.2), are

$$\begin{aligned} {[}\mathbf {M}^e]&=\int _{\varOmega ^e}[\mathbf {N}^e]^T\rho _0[\mathbf {N}^e]\,\mathrm {d}z, \end{aligned}$$

(A.6a)

$$\begin{aligned} {[}\mathbf {D}_1^e]&=\int _{\varOmega ^e}[\mathbf {N}^e]^T\rho _0z \left[ \frac{\hbox {d}\mathbf {N}^e}{\hbox {d}z}\right] \,\mathrm {d}z, \end{aligned}$$

(A.6b)

$$\begin{aligned} {[}\mathbf {D}_2^e]&=\int _{\varOmega ^e}[\mathbf {N}^e]^T\rho _0z \left[ \frac{\hbox {d}^{2}\mathbf {N}^e}{\hbox {d}z^{2}}\right] \,\mathrm {d}z, \end{aligned}$$

(A.6c)

$$\begin{aligned} {[}\mathbf {D}_3^e]&=\int _{\varOmega ^e}\left[ \frac{\hbox {d}^{2}\mathbf {N}^e}{\hbox {d}z^{2}}\right] ^T EI \left[ \frac{\hbox {d}^{2}\mathbf {N}^e}{\hbox {d}z^{2}}\right] \,\mathrm {d}z, \end{aligned}$$

(A.6d)

$$\begin{aligned} {[}\mathbf {F}^e]&=\int _{\varOmega ^e}-[\mathbf {N}^e]^T\rho _0g\,\mathrm {d}z. \end{aligned}$$

(A.6e)

We introduce the temporal discretization $t=t_k$ for $k=1,2,\ldots$ and define $\varDelta t_n=t_{n+1}-t_n$. Given $[{\mathbf {q}}_n]$, $[\dot{{\mathbf {q}}}_n]$, $[\ddot{{\mathbf {q}}}_n]$, $\gamma _n$, ${\dot{\gamma }}_n$, and $\ddot{\gamma }_n$, we seek to use Eq. (A.5) to determine $[{\mathbf {q}}_{n+1}]$, $[\dot{{\mathbf {q}}}_{n+1}]$, $[\ddot{{\mathbf {q}}}_{n+1}]$, $\gamma _{n+1}$, ${\dot{\gamma }}_{n+1}$, and $\ddot{\gamma }_{n+1}$ by way of the Newmark method. Thus, we assume

$$\begin{aligned} {\mathbf {q}}_{n+1}&={\mathbf {q}}_n+\dot{{\mathbf {q}}}_n\varDelta t_n+\frac{\varDelta t_n^2}{2}[(1-2\beta )\ddot{{\mathbf {q}}}_n+2\beta \ddot{{\mathbf {q}}}_{n+1}], \end{aligned}$$

(A.7a)

$$\begin{aligned} \dot{{\mathbf {q}}}_{n+1}&=\dot{{\mathbf {q}}}_n+[(1-\gamma )\ddot{{\mathbf {q}}}_n+\gamma \ddot{{\mathbf {q}}}_{n+1}]\varDelta t_n, \end{aligned}$$

(A.7b)

and similarly for $\gamma$. A nonlinear matrix problem for $[{\mathbf {q}}_{n+1}]$ and $\gamma _{n+1}$ results upon inserting Eq. (A.7) into Eq. (A.5). To solve, we iterate $[{\mathbf {q}}_{n+1}]$ and $\gamma _{n+1}$ until the z-domain counterpart of Eq. (2.3)$_5$ is satisfied. Then, the velocities and accelerations at $t=t_{n+1}$ can be computed from Eq. (A.7).